Monotonicity of solutions to quasilinear problems with a first-order term in half-spaces (original) (raw)

Monotonicity of solutions for some nonlocal elliptic problems in half-spaces

Calculus of Variations and Partial Differential Equations, 2017

In this paper we consider classical solutions u of the semilinear fractional problem (−∆) s u = f (u) in R N + with u = 0 in R N \ R N + , where (−∆) s , 0 < s < 1, stands for the fractional laplacian, N ≥ 2, R N + = {x = (x ′ , xN) ∈ R N : xN > 0} is the half-space and f ∈ C 1 is a given function. With no additional restriction on the function f , we show that bounded, nonnegative, nontrivial classical solutions are indeed positive in R N + and verify ∂u ∂xN > 0 in R N +. This is in contrast with previously known results for the local case s = 1, where nonnegative solutions which are not positive do exist and the monotonicity property above is not known to hold in general even for positive solutions when f (0) < 0.

Existence of Strictly Positive Solutions for Sublinear Elliptic Problems in Bounded Domains

Advanced Nonlinear Studies, 2014

Let Ω be a smooth bounded domain in RN and let m be a possibly discontinuous and unbounded function that changes sign in Ω. Let f : [0,∞) → [0,∞) be a nondecreasing continuous function such that k1ξp ≤ f (ξ) ≤ k2ξp for all ξ ≥ 0 and some k1, k2 > 0 and p ∈ (0, 1). We study existence and nonexistence of strictly positive solutions for nonlinear elliptic problems of the form −Δu = m(x) f (u) in Ω, u = 0 on ∂Ω.

Positive solutions to sublinear elliptic problems

Colloquium Mathematicum, 2018

Let L be a second order elliptic operator L with smooth coefficients defined on a domain Ω in R d , d ≥ 3, such that L1 ≤ 0. We study existence and properties of continuous solutions to the following problem (0.1) Lu = ϕ(•, u), in Ω, where Ω is a Greenian domain for L (possibly unbounded) in R d and ϕ is a nonnegative function on Ω × [0, +∞[ increasing with respect to the second variable. By means of thinness, we obtain a characterization of ϕ for which (0.1) has a nonnegative nontrivial bounded solution.

On Neumann Boundary Value Problems For Some Quasilinear Elliptic Equations

We study the role played by the indefinite weight function a(x) on the existence of positive solutions to the problem    −div (|∇u| p−2 ∇u) = λa(x)|u| p−2 u + b(x)|u| γ−2 u, x ∈ Ω, ∂u ∂n = 0, x ∈ ∂Ω, where Ω is a smooth bounded domain in R n , b changes sign, 1 < p < N, 1 < γ < Np/(N − p) and γ = p. We prove that (i) if Ω a(x) dx = 0 and b satisfies another integral condition, then there exists some λ * such that λ * Ω a(x) dx < 0 and, for λ strictly between 0 and λ * , the problem has a positive solution and (ii) if Ω a(x) dx = 0, then the problem has a positive solution for small λ provided that Ω b(x) dx < 0.

Existence of Positive Bounded Solutions of Semilinear Elliptic Problems

International Journal of Differential Equations, 2010

This paper is concerned with the existence of bounded positive solution for the semilinear elliptic problemΔu=λp(x)f(u)inΩsubject to some Dirichlet conditions, whereΩis a regular domain inℝn (n≥3)with compact boundary. The nonlinearityfis nonnegative continuous and the potentialpbelongs to some Kato classK(Ω). So we prove the existence of a positive continuous solution depending onλby the use of a potential theory approach.